Forward-Start Heston Options and Complex Moment-Generating Functions

# Pricing forward starting options in the Heston model using the Characteristic function


# Pricing forward starting options in the Heston model using the Characteristic function












I read about the pricing of forward starting options in the book "Mathematical Modeling and Computation in Finance" by A.W. Oosterlee and L.A. Grzelak, for the Heston model in particular.

Those pricing method are based on first calculating the characteristic function and then using numerical techniques involving this function to obtain the option price.

Anyhow, anyone familiar with option pricing in the Heston model knows the long backstory of how important it is to choose the right presentation for the characteristic function, meaning choosing the right branch for the complex logarithms and square roots involved.

If this is done the wrong way discontinuities will sometimes occur when the model parameters varies in the parameter space.

My question concerns this issue when we consider the forward characteristic function. Skipping some details, in this case we need to calculate the characteristic function of $$ x\left(T_1, T_2\right):=\log S\left(T_2\right)-\log S\left(T_1\right), $$ where $S$ is the underlying asset and $t_0 <T_1 <T_2$ are two timestamps and $t_0$ is the time from which we calculate the characteristic function, $\phi_x(u)$. By iterated conditioning, we have $$ \phi_x(u)=\mathbb{E}^{\mathbb{Q}}\left[\mathbb{E}^{\mathbb{Q}}\left[\mathrm{e}^{i u\left(\log S\left(T_2\right)-\log S\left(T_1\right)\right)} \mid \mathcal{F}\left(T_1\right)\right] \mid \mathcal{F}\left(t_0\right)\right] = \mathbb{E}^{\mathbb{Q}}\left[\mathrm{e}^{-i u \log S\left(T_1\right)} \mathbb{E}^{\mathbb{Q}}\left[\mathrm{e}^{i u \log S\left(T_2\right)} \mid \mathcal{F}\left(T_1\right)\right] \mid \mathcal{F}\left(t_0\right)\right]. $$ The inner expectation here is just the standard characteristic function for the logarithm of the underlying at a time $T_2$ seen from the earlier time $T_1$. By the standard theory of the Heston model, we know the form of this function: $$ \mathbb{E}^{\mathbb{Q}}\left[\mathrm{e}^{i u \log S\left(T_2\right)} \mid \mathcal{F}\left(T_1\right)\right]=\mathrm{e}^{\bar{A}(u, \tau)+ iu \log S\left(T_1\right) +\bar{C}(u, \tau) v\left(T_1\right)}, $$ Where $\tau = T_2 - T_1$, $v$ is the variance and $\bar{A}$ and $\bar{C}$ are two complex valued functions. This leads to a characteristic function of the form $$ \phi_x(u)=\mathrm{e}^{\bar{A}(u, \tau)} \mathbb{E}^{\mathbb{Q}}\left[\mathrm{e}^{\bar{C}(u, \tau) v\left(T_1\right)} \mid \mathcal{F}\left(t_0\right)\right]. $$ This expectation here is the moment generating function of $v(T_1)$ evaluated at an argument $\bar{C}(u, \tau)$. The values of $v(t)$ follows a CIR-process $$ \mathrm{d} v(t)=\kappa(\bar{v}-v(t)) \mathrm{d} t+\gamma \sqrt{v(t)} \mathrm{d} W_v(t), \quad v\left(t_0\right)=v_0, $$ for $t \geq t_0$. The previously mentioned book derives the moment generating function for this: $$ \mathbb{E}^{\mathbb{Q}}\left[\mathrm{e}^{u v(t)} \mid \mathcal{F}\left(t_0\right)\right]=\left(\frac{1}{1-2 u \bar{c}\left(t, t_0\right)}\right)^{\frac{1}{2} \delta} \exp \left(\frac{u \bar{c}\left(t, t_0\right) \bar{\kappa}\left(t, t_0\right)}{1-2 u \bar{c}\left(t, t_0\right)}\right), $$ where $$ \bar{c}\left(t, t_0\right)=\frac{\gamma^2}{4 \kappa}\left(1-\mathrm{e}^{-\kappa\left(t-t_0\right)}\right), \quad \delta=\frac{4 \kappa \bar{v}}{\gamma^2}, \quad \bar{\kappa}\left(t, t_0\right)=\frac{4 \kappa v_0 \mathrm{e}^{-\kappa\left(t-t_0\right)}}{\gamma^2\left(1-\mathrm{e}^{-\kappa\left(t-t_0\right)}\right)}. $$ Then they use this result an plug in $\bar{C}(u, \tau)$ in the previous formula for $\phi_x(u)$ to get $$ \begin{aligned} \phi_x(u) & =\mathrm{e}^{\bar{A}(u, \tau)} \mathbb{E}^{\mathbb{Q}}\left[\mathrm{e}^{\bar{C}(u, \tau) v\left(T_1\right)} \mid \mathcal{F}\left(t_0\right)\right] \\ & =\exp \left(\bar{A}(u, \tau)+\frac{\bar{C}(u, \tau) \bar{c}\left(T_1, t_0\right) \bar{\kappa}\left(T_1, t_0\right)}{1-2 \bar{C}(u, \tau) \bar{c}\left(T_1, t_0\right)}\right)\left(\frac{1}{1-2 \bar{C}(u, \tau) \bar{c}\left(T_1, t_0\right)}\right)^{\frac{1}{2} \delta}. \end{aligned} $$

My question here is how we can be sure that we can evaluate this moment generating function for the argument $\bar{C}(u, \tau)$, a complex valued function coming from the Heston model?

There is no, a priore, reason for the moment generating function to be valid for all complex arguments. For example, some arguments will cause division by zero.

Also, we raise a complex number to the power $\delta/2$. This operation involves choosing a branch of the complex logarithm. How can we be sure we don't re-introduce the same type of problems with branch cut discontinuities, as the parameters varies in the parameter space, we had for years with the standard Heston characteristic function before a proper formulation could be found?

Maybe this has been investigated somewhere or I am missing something obvious, but I would be happy to be pointed to some references or explanation then.